Continuum and soft robots can leverage complex actuator shapes to take on useful shapes while actuating only a few of their many degrees of freedom. Continuum robots that also grow increase the range of potential shapes that can be actuated and enable easier access to constrained environments. Existing models for describing the complex kinematics involved in general actuation of continuum robots rely on simulation or well-behaved stress-strain relationships, but the non-linear behavior of the thin-walled inflated-beams used in growing robots makes these techniques difficult to apply. Here we derive kinematic models of single, generally routed tendon paths on a soft pneumatic backbone of inextensible but flexible material from geometric relationships alone. This allows for forward modeling of the resulting shapes with only knowledge of the geometry of the system. We show that this model can accurately predict the shape of the whole robot body and how the model changes with actuation type. We also demonstrate the use of this kinematic model for inverse design, where actuator designs are found based on desired final robot shapes. We deploy these designed actuators on soft pneumatic growing robots to show the benefits of simultaneous growth and shape change.
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